Similarity Classes of Nilpotents
Published:
We compute the closure of the similarity classes of nilpotents of \(M_n(\mathbb C)\). This is the Gerstenhaber–Hesselink theorem.
Posts by Tags
abstract algebra
The Irrationality of \(\sqrt{2}\)
Published:
We collect ten(ish) proofs that \(\sqrt{2}\) is irrational.
algebra
Filtered Algebras
Published:
We give a criterion for a filtered algebra to be a polynomial ring.
algebraic geometry
Hilbert’s Nullstellensatz
Published:
We record a proof of the bijection of algebraic sets and radical ideals, using Hilbert’s Nullstellensatz.
automorphism
Groups With No Automorphisms
Published:
We classify the groups with trivial automorphism groups. This was a pretty tricky problem on my girlfriend’s algebra qualifying exam the other day!
category theory
The Eckmann–Hilton Argument
Published:
We describe the Eckmann–Hilton argument and apply it to topological groups.
class field theory
Dirichlet Characters as Langlands
Published:
We give an exposition of Dirichlet characters which has Langlands philosophy in mind.
field theory
Automorphisms of Fields
Published:
We describe some rigidity (or lack thereof) results for field automorphisms. This is based on another problem from my girlfriend’s algebra exam.
filters
Tychonoff’s Theorem via Filters
Published:
To give myself some practice with filters, we describe the proof of Tychonoff’s theorem via filters.
finite fields
The Sign of the Linear Action
Published:
We compute the sign of \(\operatorname{GL}_n(\mathbb F_q)\) acting on \(\mathbb F_q^{\oplus n}\).
fundamental group
The Eckmann–Hilton Argument
Published:
We describe the Eckmann–Hilton argument and apply it to topological groups.
gauss sums
Quadratic Subfields of Cyclotomic Fields
Published:
We use Gauss sums to work out the quadratic subfields of cyclotomic fields.
group cohomology
Lang’s Theorem
Published:
We give the standard proof of Lang’s theorem. This post will use some algebraic geometry.
group theory
The Eckmann–Hilton Argument
Published:
We describe the Eckmann–Hilton argument and apply it to topological groups.
The Sign of the Linear Action
Published:
We compute the sign of \(\operatorname{GL}_n(\mathbb F_q)\) acting on \(\mathbb F_q^{\oplus n}\).
Groups With No Automorphisms
Published:
We classify the groups with trivial automorphism groups. This was a pretty tricky problem on my girlfriend’s algebra qualifying exam the other day!
homotopy theory
The Homotopy Fiber of \(\mathrm{Sq}^1\)
Published:
We use the Serre spectral sequence to compute the homotopy fiber of the Steenrod square \(\mathrm{Sq}^1\colon K(\mathbb F_2,1)\to K(\mathbb F_2,2)\).
The Cohomology of \(K(\mathbb Z,3)\)
Published:
We use the Serre spectral sequence to compute the first ten cohomology groups of \(K(\mathbb Z,3)\).
l functions
Special Values of Dirichlet \(L\)-Functions
Published:
Inspired by a question of Austin Lei, we review the basic theory of special values of Dirichlet \(L\)-functions.
nets
Nets
Published:
For my own personal use, we review the basic theory of nets.
nullstellensatz
Hilbert’s Nullstellensatz
Published:
We record a proof of the bijection of algebraic sets and radical ideals, using Hilbert’s Nullstellensatz.
number theory
The Hermite–Minkowski Theorem
Published:
We give the standard proof of the Hermite–Minkowski theorem.
Minkowski Theory
Published:
We give a standard presentation of Minkowski theory.
Genus Theory
Published:
We compute the \(2\)-part of the class group of an imaginary quadratic field \(K\).
The Quadratic Class Number Formula
Published:
We work out some examples of the class number formula for imaginary quadratic fields.
Evaluation of \(L(1,\chi)\)
Published:
We compute \(\left|L(1,\chi)\right|\) for an odd primitive Dirichlet character \(\chi\pmod N\).
Quadratic Dirichlet Characters
Published:
We work out some basic facts about quadratic Dirichlet characters from the perspective of Galois theory.
Quadratic Subfields of Cyclotomic Fields
Published:
We use Gauss sums to work out the quadratic subfields of cyclotomic fields.
Dirichlet Characters as Langlands
Published:
We give an exposition of Dirichlet characters which has Langlands philosophy in mind.
Lang’s Theorem
Published:
We give the standard proof of Lang’s theorem. This post will use some algebraic geometry.
Special Values of Dirichlet \(L\)-Functions
Published:
Inspired by a question of Austin Lei, we review the basic theory of special values of Dirichlet \(L\)-functions.
The Irrationality of \(\sqrt{2}\)
Published:
We collect ten(ish) proofs that \(\sqrt{2}\) is irrational.
permutations
The Sign of the Linear Action
Published:
We compute the sign of \(\operatorname{GL}_n(\mathbb F_q)\) acting on \(\mathbb F_q^{\oplus n}\).
polynomials
Units in Polynomial Rings
Published:
We classify the units in polynomial rings.
ring theory
Hilbert’s Nullstellensatz
Published:
We record a proof of the bijection of algebraic sets and radical ideals, using Hilbert’s Nullstellensatz.
Units in Polynomial Rings
Published:
We classify the units in polynomial rings.
topology
Similarity Classes of Nilpotents
Published:
We compute the closure of the similarity classes of nilpotents of \(M_n(\mathbb C)\). This is the Gerstenhaber–Hesselink theorem.
Automorphisms of Fields
Published:
We describe some rigidity (or lack thereof) results for field automorphisms. This is based on another problem from my girlfriend’s algebra exam.
Tychonoff’s Theorem via Filters
Published:
To give myself some practice with filters, we describe the proof of Tychonoff’s theorem via filters.
Nets
Published:
For my own personal use, we review the basic theory of nets.
tychonoff
Tychonoff’s Theorem via Filters
Published:
To give myself some practice with filters, we describe the proof of Tychonoff’s theorem via filters.
units
Units in Polynomial Rings
Published:
We classify the units in polynomial rings.